Notes:An introduction to manifolds - Loring W. Tu

Chapter 1

Section 1: Smooth functions on Euclidean space

1.1: $C^\infty$ vs Analytic functions

Example:A smooth function that is not real analytic

Section 2: Tangent vectors in $\mathbb{R}^n$ as Derivations

• $$D_vf\eq\lim_{t\rightarrow 0}\left(\frac{f(c(t))-f(p)}{t}\right)\eq\frac{d}{dt}f(c(t))\Big\vert_{t\eq 0}$$

2.2: Germs of functions

• Equivalence relation yeah yeah yeah
• We define an equivalence relation on the $C^\infty$ functions defined in some neighbourhood of $p\in\mathbb{R}^n$.
• Consider the set of all ordered pairs, $(f,U)$, where $U$ is an open neighbourhood and $f:U\rightarrow\mathbb{R}$ is a $C^\infty$ (AKA: Smooth function).
  • We define $(f,U)\sim(g,V)$ if:
    • There exists an open set: $W\subseteq U\cap V$ with $p\in W$ such that:
      • $f\big\vert_W\eq g\big\vert_W$
• Germ: The equivalence class of $(f,U)$ is called the Germ of $f$ at $p$
• We write the set of all germs of $C^\infty$ functions on $\mathbb{R}^n$ at $p$ as:
  • $C^\infty_p(\mathbb{R}^n)$ or simply $C^\infty_p$ if the space is obvious.

Example

Define:

• $$f:(\mathbb{R}-\{1\})\rightarrow\mathbb{R}$$ by $f:x\mapsto\dfrac{1}{1-x}$ and
• $g:(0,1)\rightarrow\mathbb{R}$ by $$g:x\mapsto 1+\sum^\infty_{n\eq 1}x^n$$

Then $f$ and $g$ have the same germ at any point $p\in (-1,1)$

Grade: B

There is a topic that needs to be explored here

Some aspect of this work needs exploring and researching in more detail.
The message provided is:

This is an important concept

Then we just hit the concept of an algebra

Section 3: The exterior algebra of multicovectors

3.3: Multilinear functions

• $k$-linear function. A multilinear function: $f:V^k\rightarrow\mathbb{R}$.
• Future: Permutation action: Let $f$ be $k$-linear and let $\sigma\in S_k$ - the symmetric group on $k$ symbols. Then:
  • $(\sigma f)(v_1,\ldots,v_k):\eq f(v_{\sigma(1)},\ldots,v_{\sigma(k)})$
• Symmetric: $\forall\sigma\in S_k[\sigma f\eq f]$
• Alternating: $\forall\sigma\in S_k[\sigma f\eq\text{Sign}(\sigma)f]$

Notations

• $L_k(V)$ - all $k$-linear functions
• $A_k(V)$ - all alternating $k$-linear functions.

Lemma 3.11:

• If $\sigma,\tau\in S_k$ and $f$ is $k$-linear then:
  • $\tau(\sigma f)\eq(\tau\sigma)f$

3.5: The symmetrising and alternating operators

Let $f\in L_k(V)$, then:

• $$Sf:\eq \sum_{\sigma \in S_k} \sigma f$$
• $$Af:\eq \sum_{\sigma \in S_k} \text{Sign}(\sigma)\sigma f$$

Lemma 3.14:

• If $f\in L_k(V)$ is an alternating $k$-linear function already then:
  • $Af\eq (k!)f$

3.6: The tensor product

Let $f\in L_k(V)$ and $g\in L_\ell(V)$, then their tensor product is a $(k+\ell)$-linear function, $f\otimes g$ defined as follows:

• • $(f\otimes g)(v_1,\ldots,v_{k+\ell}):\eq f(v_1,\ldots,v_k)g(v_{k+1},\ldots,v_{k+\ell})$

3.7: The wedge product

Let $f\in A_k(V)$ and $g\in A_\ell(V)$, the wedge product is a product that is alternating also:

• $f\wedge g:\eq \frac{1}{k!\ell !}A(f\otimes g)$, or explicitly:
• $$f\wedge g(v_1,\ldots,v_{k+\ell})\eq\frac{1}{k!\ell!}\sum_{\sigma\in S_{k+\ell} } \text{Sign}(\sigma)f(v_{\sigma(1)},\ldots, v_{\sigma(k)})g(v_{\sigma(k+1)},\ldots,v_{\sigma(k+\ell)})$$

This is obviously alternating.

Suppose that $f(v_1,v_2)g(\text{whatever})$ is a term, then so is $-f(v_2,v_1)g(\text{whatever})$ say too.

Remember $f$ is alternating by definition, that means:

• $f(v_2,v_1)\eq -f(v_1,v_2)$

So we really have $2f(v_1,v_2)g(\text{whatever})$ in the term. There are a lot of redundancies.

Definition:

• A permutation, $\sigma\in S_{k+\ell}$ is a $(k,\ell)$-shuffle if:
  • $\sigma(1)<\sigma(2)<\cdots<\sigma(k-1)<\sigma(k)$ and $\sigma(k+1)<\sigma(k+2)<\cdots<\sigma(k+\ell-1)<\sigma(k+\ell)$

Now we may re-write $f\wedge g$ as:

• $$(f\wedge g)\eq\sum_{\sigma\ :\ (k,\ell)\text{-shuffle} } \text{Sign}(\sigma)f(v_{\sigma(1)},\ldots,v_{\sigma(k)})g(v_{\sigma(k+1)},\ldots,v_{\sigma(k+\ell)})$$

Caveat:All of this is VERY informal... there needs to be proof.... but I'll go along

Section 4: Differential Forms on $\mathbb{R}^n$

4.1: Differential $1$-forms and the differential of a function

• Cotangent space: to $\mathbb{R}^n$ at $p\in\mathbb{R}^n$ is denoted by $T^*_p(\mathbb{R}^n)$ is defined to be:
  • $(T_p(\mathbb{R}^n))^*$ - or $(T_p(\mathbb{R}^n))^\vee$ in this author's notation - of the tangent space $T_p(\mathbb{R}^n)$